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Solutions to Sobolev Supercritical Nonlinear Schrodinger Equations on an Annulus via a Hopf Reduction Method

Jian Liang, Hua-Yang Wang

TL;DR

The paper tackles normalized, positive solutions to Sobolev supercritical nonlinear Schrödinger equations on an annulus by employing a Hopf fibration–based reduction to a lower-dimensional weighted problem in dimensions M=3,5,9 (corresponding to N=4,8,16). It establishes a mass-threshold dichotomy: for subcritical/critical nonlinearities there exists a global minimizer for all masses, while in the supercritical regime there are two positive solutions (a local minimizer and a mountain-pass solution) for small prescribed mass. The results are then translated back to the original higher-dimensional setting via reverse reduction, and the paper develops a monotonicity-trick framework together with a blow-up analysis to guarantee the mountain-pass solution. These findings advance understanding of Sobolev supercritical NLS with mass constraints on topologically nontrivial domains and highlight the power of symmetry-based reductions in nonlinear PDE variational problems.

Abstract

This paper investigates the existence of positive solutions with a prescribed mass for nonlinear Schrodinger equations on an annulus, possibly in the Sobolev supercritical regime. A reduction method based on the Hopf fibration is used to transform the problem into a lower-dimensional one. We obtain a new mass critical threshold and we show that in the new mass subcritical or critical regimes there exists a positive solution which corresponds to a global minimizer, while in the mass supercritical regime, there exists two positive solutions which correspond to a local minimizer and a mountain pass solution respectively. Some other problems are also discussed in this paper.

Solutions to Sobolev Supercritical Nonlinear Schrodinger Equations on an Annulus via a Hopf Reduction Method

TL;DR

The paper tackles normalized, positive solutions to Sobolev supercritical nonlinear Schrödinger equations on an annulus by employing a Hopf fibration–based reduction to a lower-dimensional weighted problem in dimensions M=3,5,9 (corresponding to N=4,8,16). It establishes a mass-threshold dichotomy: for subcritical/critical nonlinearities there exists a global minimizer for all masses, while in the supercritical regime there are two positive solutions (a local minimizer and a mountain-pass solution) for small prescribed mass. The results are then translated back to the original higher-dimensional setting via reverse reduction, and the paper develops a monotonicity-trick framework together with a blow-up analysis to guarantee the mountain-pass solution. These findings advance understanding of Sobolev supercritical NLS with mass constraints on topologically nontrivial domains and highlight the power of symmetry-based reductions in nonlinear PDE variational problems.

Abstract

This paper investigates the existence of positive solutions with a prescribed mass for nonlinear Schrodinger equations on an annulus, possibly in the Sobolev supercritical regime. A reduction method based on the Hopf fibration is used to transform the problem into a lower-dimensional one. We obtain a new mass critical threshold and we show that in the new mass subcritical or critical regimes there exists a positive solution which corresponds to a global minimizer, while in the mass supercritical regime, there exists two positive solutions which correspond to a local minimizer and a mountain pass solution respectively. Some other problems are also discussed in this paper.

Paper Structure

This paper contains 15 sections, 13 theorems, 164 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivations and backgrounds
  3. Main results
  4. Plan of the paper
  5. Preliminaries
  6. The Hopf reduction method
  7. Gagliardo-Nirenberg inequality
  8. A global minimizer
  9. A local minimizer and a mountain pass solution
  10. Existence of a local minimizer
  11. The mountain pass geometry
  12. The abstract framework of monotonicity trick
  13. The blow-up analysis
  14. Comments and open problems
  15. The reduction process

Key Result

Theorem 1.1

Let $M \geq 3$. Assume that $f$ satisfies the assumptions ($\mathbf{A_1}$) and ($\mathbf{A_2}$). Denote the mass critical exponent in dimension $M$ by $2^{\#}_M = 2 + \frac{4}{M}$, then the following conclusions hold:

Figures (1)

  • Figure 1: The domain $\Omega$ in the $(r,s)$-plane.

Theorems & Definitions (33)

  • Remark 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.2
  • Remark 1.3
  • Remark 2.1
  • Lemma 3.1
  • proof
  • Remark 3.1
  • Definition 3.1
  • ...and 23 more